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Arithmetic (from the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
ἀριθμός ''arithmos'', '
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
' and τική έχνη ''tiké échne', '
art Art is a diverse range of (products of) human activities Humans (''Homo sapiens'') are the most populous and widespread species of primates, characterized by bipedality, opposable thumbs, hairlessness, and intelligence allowing the use ...

art
' or '
craft A craft or trade is a pastime or an occupation that requires particular skills and knowledge of skilled work. In a historical sense, particularly the Middle Ages In the history of Europe The history of Europe concerns itself wit ...
') is a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
that consists of the study of
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
s, especially the properties of the traditional
operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...
on them—
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

subtraction
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
,
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
,
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
and extraction of
roots A root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a lar ...
. Arithmetic is an elementary part of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, and number theory is considered to be one of the top-level divisions of modern mathematics, along with
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
,
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, and
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
. The terms ''arithmetic'' and ''higher arithmetic'' were used until the beginning of the 20th century as synonyms for ''number theory'', and are sometimes still used to refer to a wider part of number theory.


History

The prehistory of arithmetic is limited to a small number of artifacts, which may indicate the conception of addition and subtraction, the best-known being the
Ishango bone The Ishango bone, discovered at the "Fisherman Settlement" of Ishango in the Democratic Republic of Congo, is a bone toolIn archaeology Archaeology or archeology is the study of human activity through the recovery and analysis of material c ...

Ishango bone
from
central Africa Central Africa is a subregion A subregion is a part of a larger region In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhab ...

central Africa
, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed. The earliest written records indicate the
Egyptians Egyptians ( arz, المصريين, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group of people originating from the country of Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a spanning t ...
and
Babylonians Babylonia () was an ancient Ancient history is the aggregate of past eventsWordNet Search – ...
used all the
elementary arithmetic Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is sig ...
operations as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
strongly influence the complexity of the methods. The hieroglyphic system for
Egyptian numerals The system of ancient Egyptian numerals was used in Ancient Egypt Ancient Egypt was a civilization of Ancient history, ancient North Africa, concentrated along the lower reaches of the Nile, Nile River, situated in the place that is now th ...
, like the later
Roman numerals Roman numerals are a that originated in and remained the usual way of writing numbers throughout Europe well into the . Numbers in this system are represented by combinations of letters from the . Modern style uses seven symbols, each with a ...
, descended from
tally marks frame, Brahmi numerals (lower row) in India in the 1st century CE. Note the similarity of the first three numerals to the Chinese characters for one through three (一 二 三), plus the resemblance of both sets of numerals to horizontal tally ma ...

tally marks
used for counting. In both cases, this origin resulted in values that used a
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
base, but did not include
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
. Complex calculations with Roman numerals required the assistance of a
counting boardImage:Rechentisch.png, Rechentisch/Counting board (engraving probably from Strasbourg) The counting board is the precursor of the abacus, and the earliest known form of a counting device (excluding fingers and other very simple methods). Counting bo ...

counting board
(or the
Roman abacus The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those used by the Greeks and Babylonians. It was the first portable calculating device for Engineer, engineers, Merchant, ...
) to obtain the results. Early number systems that included positional notation were not decimal, including the
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
(base 60) system for
Babylonian numerals 450px, Babylonian cuneiform numerals Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system, script that was used to write several languages of the Ancient Nea ...

Babylonian numerals
, and the
vigesimal A vigesimal () or base-20 (base-score) numeral system is based on 20 (number), twenty (in the same way in which the decimal, decimal numeral system is based on 10 (number), ten). ''wikt:vigesimal#English, Vigesimal'' is derived from the Latin adje ...
(base 20) system that defined
Maya numerals The Mayan numeral system was the system to represent number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. ...

Maya numerals
. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation. The continuous historical development of modern arithmetic starts with the
Hellenistic civilization The Hellenistic period covers the period of Mediterranean history between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in 31 BC and the conquest of Ptolemaic ...
of ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works of
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example,
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Ancient Greek, Greek. He was born in Gerasa ...
summarized the viewpoint of the earlier
Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philos ...
approach to numbers, and their relationships to each other, in his ''
Introduction to Arithmetic The book ''Introduction to Arithmetic'' ( grc-gre, Ἀριθμητικὴ εἰσαγωγή, ''Arithmetike eisagoge'') is the only extant work on mathematics by Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 ...
''.
Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or ear ...
were used by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
,
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the autho ...
and others in a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
not very different from the modern notation. The ancient Greeks lacked a symbol for zero until the Hellenistic period, and they used three separate sets of symbols as digits: one set for the units place, one for the tens place, and one for the hundreds. For the thousands place, they would reuse the symbols for the units place, and so on. Their addition algorithm was identical to the modern method, and their multiplication algorithm was only slightly different. Their long division algorithm was the same, and the digit-by-digit square root algorithm, popularly used as recently as the 20th century, was known to Archimedes (who may have invented it). He preferred it to Hero's method of successive approximation because, once computed, a digit does not change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a fractional part, such as 546.934, they used negative powers of 60—instead of negative powers of 10 for the fractional part 0.934. The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra. The ancient Chinese used a positional notation similar to that of the Greeks. Since they also lacked a symbol for
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
, they had one set of symbols for the units place, and a second set for the tens place. For the hundreds place, they then reused the symbols for the units place, and so on. Their symbols were based on the ancient
counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia East Asia is the eastern region of Asia, which is defined in both Geography, geographical and culture, et ...
. The exact time where the Chinese started calculating with positional representation is unknown, though it is known that the adoption started before 400 BC. The ancient Chinese were the first to meaningfully discover, understand, and apply negative numbers. This is explained in the ''
Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained () ...
'' (''Jiuzhang Suanshu''), which was written by
Liu Hui Liu Hui () was a Chinese mathematician and writer who lived in the state of Cao Wei Wei (220–266), also known as Cao Wei or Former Wei, was one of the three major states that competed for supremacy over China in the Three Kingdoms perio ...
dated back to 2nd century BC. The gradual development of the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...
independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base, and the use of a digit representing
0
0
. This allowed the system to consistently represent both large and small integers—an approach which eventually replaced all other systems. In the early the Indian mathematician
Aryabhata Aryabhata (, ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century,
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

Brahmagupta
established the use of 0 as a separate number, and determined the results for multiplication, division, addition and subtraction of zero and all other numbers—except for the result of
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. His contemporary, the
SyriacSyriac may refer to: *Syriac language, a dialect of Middle Aramaic * Syriac alphabet ** Syriac (Unicode block) ** Syriac Supplement * Neo-Aramaic languages also known as Syriac in most native vernaculars * Syriac Christianity, the churches using Syr ...

Syriac
bishop Severus Sebokht (650 AD) said, "Indians possess a method of calculation that no word can praise enough. Their rational system of mathematics, or of their method of calculation. I mean the system using nine symbols." The Arabs also learned this new method and called it ''hesab''. Although the
Codex Vigilanus The ''Codex Vigilanus'' or ''Codex Albeldensis'' (Spanish: ''Códice Vigilano'' or ''Albeldense'') is an illuminated compilation of various historical documents accounting for a period extending from antiquity to the 10th century in Hispania ...
described an early form of Arabic numerals (omitting 0) by 976 AD, Leonardo of Pisa (
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of I ...

Fibonacci
) was primarily responsible for spreading their use throughout Europe after the publication of his book ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
'' in 1202. He wrote, "The method of the Indians (Latin ''Modus Indorum'') surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
". In the Middle Ages, arithmetic was one of the seven
liberal arts Liberal arts education (from Latin "free" and "art or principled practice") is the traditional academic program in Western higher education. ''Liberal arts'' takes the term ''Art (skill), art'' in the sense of a learned skill rather than spec ...
taught in universities. The flourishing of
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
in the
medieval In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...

medieval
Islamic Islam (; ar, اَلْإِسْلَامُ, al-’Islām, "submission
o God Oh God may refer to: * An exclamation; similar to "oh no", "oh yes", "oh my", "aw goodness", "ah gosh", "ah gawd"; see interjection An interjection is a word or expression that occurs as an utterance on its own and expresses a spontaneous feeling ...
) is an Abrahamic religions, Abrahamic monotheistic religion teaching that Muhammad is a Muhammad in Islam, messenger of God.Peters, F. E. 2009. "Allāh." In , ed ...
world, and also in
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in ...

Renaissance
Europe Europe is a continent A continent is any of several large landmass A landmass, or land mass, is a large region In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of scienc ...

Europe
, was an outgrowth of the enormous simplification of
computation Computation is any type of calculation that includes both arithmetical and non-arithmetical steps and which follows a well-defined model (e.g. an algorithm). Mechanical or electronic devices (or, History of computing hardware, historically, peop ...

computation
through
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
notation. Various types of tools have been invented and widely used to assist in numeric calculations. Before Renaissance, they were various types of
abaci The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool that has been in use since ancient times and is still in use today. It was used in the ancient Near East The ancient Near East was the home of ea ...
. More recent examples include
slide rule The slide rule is a mechanical . The slide rule is used primarily for and and for functions such as , , s, and . They are not designed for addition or subtraction which was usually performed manually, with used to keep track of the magnitude ...

slide rule
s,
nomogram A nomogram (from Greek νόμος ''nomos'', "law" and γραμμή ''grammē'', "line"), also called a nomograph, alignment chart, or abaque, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphi ...

nomogram
s and
mechanical calculator A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically. Most mechanical calculators were comparable in size to small desktop computers and have been rendered obso ...
s, such as
Pascal's calculator A Pascaline signed by Pascal in 1652 Top view and overview of the entire mechanism Pascal's calculator (also known as the arithmetic machine or Pascaline) is a mechanical calculator invented by Blaise Pascal in the mid 17th century. Pascal was le ...
. At present, they have been supplanted by electronic
calculator An electronic calculator is typically a portable device used to perform s, ranging from basic to complex . The first calculator was created in the early 1960s. Pocket-sized devices became available in the 1970s, especially after the , the f ...

calculator
s and
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

computer
s.


Arithmetic operations

The basic arithmetic operations are addition, subtraction, multiplication and division, although arithmetic also includes more advanced operations, such as manipulations of
percentage In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

percentage
s,
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
s,
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
,
logarithmic function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s, and even
trigonometric function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

trigonometric function
s, in the same vein as logarithms (
prosthaphaeresis Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorit ...
). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, either—most common, together with
infix notation Infix notation is the notation commonly used in arithmetical and logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ) ...
—explicitly using parentheses and relying on , or using a
prefix A prefix is an affix In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) ...
or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
) can be performed, and where these four operations obey the usual laws (including distributivity), is called a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
.


Addition

Addition, denoted by the symbol +, is the most basic operation of arithmetic. In its simple form, addition combines two numbers, the ''addends'' or ''terms'', into a single number, the of the numbers (such as or ). Adding finitely many numbers can be viewed as repeated simple addition; this procedure is known as
summation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

summation
, a term also used to denote the definition for "adding infinitely many numbers" in an
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. Repeated addition of the number  is the most basic form of
counting Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...
; the result of adding is usually called the
successor Successor is someone who, or something which succeeds or comes after (see success (disambiguation), success and Succession (disambiguation), succession) Film and TV * The Successor (film), ''The Successor'' (film), a 1996 film including Laura Girli ...
of the original number. Addition is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
and
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, so the order in which finitely many terms are added does not matter. The has the property that, when added to any number, it yields that same number; so, it is the
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of addition, or the
additive identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. For every number , there is a number denoted , called the '' opposite'' of , such that and . So, the opposite of is the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
of with respect to addition, or the
additive inverse In mathematics, the additive inverse of a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be ...
of . For example, the opposite of is , since . Addition can also be interpreted geometrically, as in the following example. If we have two sticks of lengths ''2'' and ''5'', then, if the sticks are aligned one after the other, the length of the combined stick becomes ''7'', since .


Subtraction

Subtraction, denoted by the symbol -, is the inverse operation to addition. Subtraction finds the ''difference'' between two numbers, the ''minuend'' minus the ''subtrahend'': Resorting to the previously established addition, this is to say that the difference is the number that, when added to the subtrahend, results in the minuend: For positive arguments and holds: :If the minuend is larger than the subtrahend, the difference is positive. :If the minuend is smaller than the subtrahend, the difference is negative. In any case, if minuend and subtrahend are equal, the difference Subtraction is neither
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
nor
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. For that reason, the construction of this inverse operation in modern algebra is often discarded in favor of introducing the concept of inverse elements (as sketched under ), where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, . The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial)
unary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved. For any representation of numbers, there are methods for calculating results, some of which are particularly advantageous in exploiting procedures, existing for one operation, by small alterations also for others. For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of
two's complement Two's complement is a mathematical operation In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (called ''operands'') to a well-defined output value. The number of operands is the arity of the ...
for representing the additive inverses, which is extremely easy to implement in hardware (
negation In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...
). The trade-off is the halving of the number range for a fixed word length. A formerly wide spread method to achieve a correct change amount, knowing the due and given amounts, is the ''counting up method'', which does not explicitly generate the value of the difference. Suppose an amount ''P'' is given in order to pay the required amount ''Q'', with ''P'' greater than ''Q''. Rather than explicitly performing the subtraction ''P'' − ''Q'' = ''C'' and counting out that amount ''C'' in change, money is counted out starting with the successor of ''Q'', and continuing in the steps of the currency, until ''P'' is reached. Although the amount counted out must equal the result of the subtraction ''P'' − ''Q'', the subtraction was never really done and the value of ''P'' − ''Q'' is not supplied by this method.


Multiplication

Multiplication, denoted by the symbols \times or \cdot, is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the ''product''. The two original numbers are called the ''multiplier'' and the ''multiplicand'', mostly both are simply called ''factors''. Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number greater than 1, say ''x'', is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where ''x'' was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards 0, in such a way that 1 goes to the multiplicand. Another view on multiplication of integer numbers (extendable to rationals but not very accessible for real numbers) is by considering it as repeated addition. For example. corresponds to either adding times a , or times a , giving the same result. There are different opinions on the advantageousness of these paradigmata in math education. Multiplication is commutative and associative; further, it is distributive over addition and subtraction. The
multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
is 1, since multiplying any number by 1 yields that same number. The
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

multiplicative inverse
for any number except  is the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...
of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity .  is the only number without a multiplicative inverse, and the result of multiplying any number and is again One says that is not contained in the multiplicative
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
of the numbers. The product of ''a'' and ''b'' is written as or . When ''a'' or ''b'' are expressions not written simply with digits, it is also written by simple juxtaposition: ''ab''. In computer programming languages and software packages (in which one can only use characters normally found on a keyboard), it is often written with an asterisk: a * b. Algorithms implementing the operation of multiplication for various representations of numbers are by far more costly and laborious than those for addition. Those accessible for manual computation either rely on breaking down the factors to single place values and applying repeated addition, or on employing
tables Table may refer to: * Table (information) A table is an arrangement of information or data, typically in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data analysis. Tables ap ...
or slide rules, thereby mapping multiplication to addition and vice versa. These methods are outdated and are gradually replaced by mobile devices. Computers utilize diverse sophisticated and highly optimized algorithms, to implement multiplication and division for the various number formats supported in their system.


Division

Division, denoted by the symbols \div or /, is essentially the inverse operation to multiplication. Division finds the ''quotient'' of two numbers, the ''dividend'' divided by the ''divisor''. Any dividend divided by zero is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than or equal to 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend. Division is neither commutative nor associative. So as explained in , the construction of the division in modern algebra is discarded in favor of constructing the inverse elements with respect to multiplication, as introduced in . Hence division is the multiplication of the dividend with the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...

reciprocal
of the divisor as factors, that is, Within the natural numbers, there is also a different but related notion called
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
, which outputs two numbers after "dividing" a natural (numerator) by a natural (denominator): first a natural (quotient), and second a natural (remainder) such that and In some contexts, including computer programming and advanced arithmetic, division is extended with another output for the remainder. This is often treated as a separate operation, the
Modulo operation In computing, the modulo operation returns the remainder or signed remainder of a Division (mathematics), division, after one number is divided by another (called the ''modular arithmetic, modulus'' of the operation). Given two positive numbers a ...
, denoted by the symbol % or the word mod, though sometimes a second output for one "divmod" operation. In either case,
Modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
has a variety of use cases. Different implementations of division (floored, truncated, Euclidean, etc) correspond with different implementations of modulus.


Fundamental theorem of arithmetic

The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (a representation of a number as the product of prime factors), excluding the order of the factors. For example, 252 only has one prime factorization: :252 = 2 × 3 × 7
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a colle ...
first introduced this theorem, and gave a partial proof (which is called
Euclid's lemma In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A na ...
). The fundamental theorem of arithmetic was first proven by
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Carl Friedrich Gauss
. The fundamental theorem of arithmetic is one of the reasons why 1 is not considered a prime number. Other reasons include the
sieve of Eratosthenes In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

sieve of Eratosthenes
, and the definition of a prime number itself (a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.).


Decimal arithmetic

Decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
refers exclusively, in common use, to the written
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
employing
arabic numerals Arabic numerals are the ten numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a Positional notation, positional numeral sy ...

arabic numerals
as the digits for a
radix In a positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally, a positional system is a numeral system in which the contribution ...

radix
10 ("decimal")
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
; however, any
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
based on powers of 10, e.g.,
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
,
Cyrillic , bg, кирилица , mk, кирилица , russian: кириллица , sr, ћирилица, uk, кирилиця , fam1 = Egyptian hieroglyphs Egyptian hieroglyphs () were the formal writing system A writing system is ...
,
Roman Roman or Romans most often refers to: *Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Laz ...
, or
Chinese numerals Chinese numerals are words and characters used to denote number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has be ...
may conceptually be described as "decimal notation" or "decimal representation". Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

Brahmagupta
of India. This was known during medieval Europe as "Modus Indoram" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
s using the same symbol for the different
orders of magnitude An order of magnitude is an approximation of the logarithm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...
(e.g., the "ones place", "tens place", "hundreds place") and, with a
radix pointIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, using those same symbols to represent
fractions A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, ...
(e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10−1) plus 6 hundredths (10−2). The concept of as a number comparable to the other basic digits is essential to this notation, as is the concept of 0's use as a placeholder, and as is the definition of multiplication and addition with 0. The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the
Jain Jainism (), traditionally known as ''Jain Dharma'', is an ancient Indian religion Indian religions, sometimes also termed Dharmic religions or Indic religions, are the religions that originated in the Indian subcontinent. These religion ...

Jain
text from
India India, officially the Republic of India (Hindi Hindi (Devanagari: , हिंदी, ISO 15919, ISO: ), or more precisely Modern Standard Hindi (Devanagari: , ISO 15919, ISO: ), is an Indo-Aryan language spoken chiefly in Hindi Belt, ...

India
entitled the '' Lokavibhâga'', dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into
Europe Europe is a continent A continent is any of several large landmass A landmass, or land mass, is a large region In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of scienc ...

Europe
by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of I ...

Fibonacci
using the Hindu–Arabic numeral system.
Algorism Algorism is the technique of performing basic arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Anci ...
comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value 9, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one (if not zero). This adjustment is termed a ''carry'' of the value 1. The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds 9, the ''carry'' adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from (). Additional steps define the final result. Similar techniques exist for subtraction and division. The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the
exponent Exponentiation is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
for the
radix In a positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally, a positional system is a numeral system in which the contribution ...

radix
(base) of 10 increases by 1 (to the left) or decreases by 1 (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10''n'' with
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
 ''n''. The list of values corresponding to all possible positions for a single digit is written Repeated multiplication of any value in this list by 10 produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as closed under multiplication. It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
.


Compound unit arithmetic

Compound unit arithmetic is the application of arithmetic operations to
mixed radix Mixed radix numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is th ...
quantities such as feet and inches; gallons and pints; pounds, shillings and pence; and so on. Before decimal-based systems of money and units of measure, compound unit arithmetic was widely used in commerce and industry.


Basic arithmetic operations

The techniques used in compound unit arithmetic were developed over many centuries and are well documented in many textbooks in many different languages. In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions: * Reduction, in which a compound quantity is reduced to a single quantity—for example, conversion of a distance expressed in yards, feet and inches to one expressed in inches. * Expansion, the
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
to reduction, is the conversion of a quantity that is expressed as a single unit of measure to a compound unit, such as expanding 24 oz to . * Normalization is the conversion of a set of compound units to a standard form—for example, rewriting "" as "". Knowledge of the relationship between the various units of measure, their multiples and their submultiples forms an essential part of compound unit arithmetic.


Principles of compound unit arithmetic

There are two basic approaches to compound unit arithmetic: * Reduction–expansion method where all the compound unit variables are reduced to single unit variables, the calculation performed and the result expanded back to compound units. This approach is suited for automated calculations. A typical example is the handling of time by
Microsoft Excel Microsoft Excel is a spreadsheet A spreadsheet is a computer application for organization, analysis, and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program ope ...
where all time intervals are processed internally as days and decimal fractions of a day. * On-going normalization method in which each unit is treated separately and the problem is continuously normalized as the solution develops. This approach, which is widely described in classical texts, is best suited for manual calculations. An example of the ongoing normalization method as applied to addition is shown below. The addition operation is carried out from right to left; in this case, pence are processed first, then shillings followed by pounds. The numbers below the "answer line" are intermediate results. The total in the pence column is 25. Since there are 12 pennies in a shilling, 25 is divided by 12 to give 2 with a remainder of 1. The value "1" is then written to the answer row and the value "2" carried forward to the shillings column. This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column. The intermediate total is divided by 20 as there are 20 shillings in a pound. The pound column is then processed, but as pounds are the largest unit that is being considered, no values are carried forward from the pounds column. For the sake of simplicity, the example chosen did not have farthings.


Operations in practice

During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pennies and farthings, and ready reckoners, which are books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound". The cumbersome nature of compound unit arithmetic has been recognized for many years—in 1586, the Flemish mathematician
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
published a small pamphlet called '''' ("the tenth") in which he declared the universal introduction of decimal coinage, measures, and weights to be merely a question of time. In the modern era, many conversion programs, such as that included in the Microsoft Windows 7 operating system calculator, display compound units in a reduced decimal format rather than using an expanded format (e.g. "2.5 ft" is displayed rather than ).


Number theory

Until the 19th century, ''number theory'' was a synonym of "arithmetic". The addressed problems were directly related to the basic operations and concerned
primality A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
,
divisibility In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, and the solution of equations in integers, such as
Fermat's last theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
. It appeared that most of these problems, although very elementary to state, are very difficult and may not be solved without very deep mathematics involving concepts and methods from many other branches of mathematics. This led to new branches of number theory such as
analytic number theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
,
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
,
Diophantine geometry In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be wr ...
and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem is a typical example of the necessity of sophisticated methods, which go far beyond the classical methods of arithmetic, for solving problems that can be stated in elementary arithmetic.


Arithmetic in education

Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers,
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s,
fractions A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, ...
, and
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
s (using the decimal place-value system). This study is sometimes known as algorism. The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics. Also, arithmetic was used by Ulama, Islamic Scholars in order to teach application of the rulings related to Zakat and Islamic inheritance jurisprudence, Irth. This was done in a book entitled ''The Best of Arithmetic'' by Abd-al-Fattah-al-Dumyati. The book begins with the foundations of mathematics and proceeds to its application in the later chapters.


See also

* * Lists of mathematics topics * Outline of arithmetic * Slide rule


Related topics

* Addition of natural numbers * Additive inverse * Arithmetic coding * Arithmetic mean * Arithmetic number * Arithmetic progression * Arithmetic properties * Associativity * Commutativity * Distributivity * Elementary arithmetic * Finite field arithmetic * Geometric progression * Integer * List of important publications in mathematics * Mental calculation * Number line


Notes


References

* Cunnington, Susan, ''The Story of Arithmetic: A Short History of Its Origin and Development'', Swan Sonnenschein, London, 1904 * Leonard Eugene Dickson, Dickson, Leonard Eugene, ''History of the Theory of Numbers'' (3 volumes), reprints: Carnegie Institute of Washington, Washington, 1932; Chelsea, New York, 1952, 1966 * Leonhard Euler, Euler, Leonhard,
Elements of Algebra
', Tarquin Press, 2007 * Henry Burchard Fine, Fine, Henry Burchard (1858–1928), ''The Number System of Algebra Treated Theoretically and Historically'', Leach, Shewell & Sanborn, Boston, 1891 * Louis Charles Karpinski, Karpinski, Louis Charles (1878–1956), ''The History of Arithmetic'', Rand McNally, Chicago, 1925; reprint: Russell & Russell, New York, 1965 * Øystein Ore, Ore, Øystein, ''Number Theory and Its History'', McGraw–Hill, New York, 1948 * André Weil, Weil, André, ''Number Theory: An Approach through History'', Birkhauser, Boston, 1984; reviewed: Mathematical Reviews 85c:01004


External links


MathWorld article about arithmetic
* :s:The New Student's Reference Work/Arithmetic, The New Student's Reference Work/Arithmetic (historical)
The Great Calculation According to the Indians, of Maximus Planudes
– an early Western work on arithmetic a
Convergence
* {{Authority control Arithmetic, Mathematics education